Unit 5: Analytical Geometry and Linear Relations

Linear Relation: A relation which a single straight line can be drawn through every data point and the first differences are constant
Non - Linear Relation: A single smooth curve can be drawn through every data point and the first differences are not constant

Slope and Equation of Line

Slope: The measure of the steepness of a line - rise / run or the rate of change
Slope Formula: \(`m = \dfrac{y_2 - y_1}{x_2 - x_1}`\)
Standard Form: \(`ax + by + c = 0, a \isin \mathbb{Z}, b \isin \mathbb{Z}, c \isin \mathbb{Z}`\) (must be integers and \(`a`\) must be positive)
Y-intercept Form: \(`y = mx + b`\)
Point-slope Form: \(`y_2-y_1 = m(x_2-x_1)`\)
The slope of a vertical lines is undefined
The slope of a horizontal line is 0
Parallel lines have the same slope
Perpendicular slopes are negative reciprocals

A relation can be described using
1. Table of Values (see below)
2. Equations \(`(y = 3x + 5)`\)
3. Graphs (Graphing the equation)
4. Words
When digging into the earth, the temperature rises according to the
following linear equation: \(`t = 15 + 0.01 h`\). \(`t`\) is the increase in temperature in
degrees and \(`h`\) is the depth in meters.

Parallel: 2 lines with the same slope
Perpendicular: 2 lines with slopes that are the negative reciprocal to the other. They form a 90 degree angle where they meet.
Domain: The ordered set of all possible values of the independent variable \(`x`\).
Range: The ordered set of all possible values of the dependent variable \(`y`\).
Continous Data: A data set that can be broken into smaller parts. This is represented by a Solid line.
Discrete Data: A data set that cannot be broken into smaller parts. This is represented by a Dashed line.
First Difference: the difference between 2 consecutive y values in a table of values which the difference between the x-values are constant.
Collinear Points: points that line on the same straight line

Independent Variable: A Variable in a relation which the values can be chosen or isn’t affected by anything.
Dependent Varaible: A Variable in a relation which is dependent on the independent variable.

Interpolation: Data inside the given data set range.
Extrapolation: Data outside the data set range.
Line of Best Fit: A line that goes through as many points as possible, and the points are the closest on either side of the line,
and it represents the trend of a graph.
Coefficient of Correlation: The value that indicates the strength of two variables in a relation. 1 is the strongest and 0 is the weakest.
Partial Variation: A Variation that represents a relation in which one variable is a multiple of the other plus a costant term.

Direction is always referring to:

A scatterplot graph is there to show the relation between two variables in a table of values.
A line of best fit is a straight line that describes the relation between two variables.
If you are drawing a line of best fit, try to use as many data points, have an equal amount of points onto and under the line of best fit, and keep it as a straight line.

To find first differences or any points on the line, you can use a table of values

Label your graph correctly, the scales/scaling and always the independent variable on the x-axis and the dependent variable on y-axis
Draw your Line of Best Fit correctly
Read the word problems carefully, and make sure you understand it when graphing things
Sometimes its better not to draw the shape, as it might cloud your judgement (personal exprience)
Label your lines